1 / 11

Mathematical Induction

Mathematical Induction. Composing Stamps: Example 3. Assume that we are given an infinite supply of stamps of two different denominations, 3 cents and and 5 cents. Prove using mathematical induction that it is possible to make up stamps of any value  8 cents. Outline of Proof. Illustration

hakan
Télécharger la présentation

Mathematical Induction

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Mathematical Induction Composing Stamps: Example 3 Assume that we are given an infinite supply of stamps of two different denominations, 3 cents and and 5 cents. Prove using mathematical induction that it is possible to make up stamps of any value  8 cents. Outline of Proof Illustration 8 = 3 + 5 9 = 3 + 3 + 3 = 3 * 3 10 = 5 + 5 = 2 * 5 11 = 1 * 5 + 2 * 3 Basis: We can make an 8 cents stamp by using one 3-cents and one 5-cents stamp. Inductive hypothesis: Assume that we can make stamps of values k = 8, 9, 10, … n. Inductive step: Show that we can compose a stamp of value n + 1.

  2. Composing Stamps – Actual Proof Inductive step: Show that we can compose a stamp of value n + 1. In composing n, several (or none) 3-cents and several (or none)5-cents stamps have been used. To go from n to n + 1, we consider two cases. Case 1 : If there is at least one 5-cents stamp in the collection, replace it by two 3-cents stamps. This gives us stamps of value n + 1 cents. Case 2 : Suppose that the current collection uses only 3-cents stamps. Since n  8, there must three 3-cents stamps in the collection. Replace these three stamps by two 5-cents stamps. This gives us a stamp of value n + 1 cents. The proof is complete. Another Proof

  3. Another Proof Another Proof For all n > 8, n can be expressed as a linear combination of 3 and 5, that is n = 3 * p + 5 * q for n >= 8 n + 1 = 3 * (p + 2 ) + 5 * (q – 1) if q > 1 n + 1 = 3 * ( p – 3 ) + 5 * (q + 2 ) if p > 3

  4. Implication In propositional logic "if p then q" is written as p  q and read as "p implies q". p q p  q same as p  q F F T F T T T F F T T T Review Propositional Logic – Page 1 How to remember this definition? Implication is false only when the premise is true and the consequence is false.

  5. Bi-conditional p  q means p iff q p q p  q same as F F T F T F T F F T T T Propositional Logic – Page 2 How to remember this definition? Bi-conditional is a like a magnitude comparator, it is true when both inputs are identical.

  6. More Terminology Tautology A logical expression that is always true, regardless of what truth values are assigned to its statement variables, is called a tautology. The statement p p is a tautology. Contradiction A logical expression that is always false, regardless of what truth values are assigned to its statement variables, is called a contradiction. The statement p p is a contradiction. Theorem If A and B are logical statements and if the statements A and A  B are true, then the statement B is true.

  7. Simple Logic Proofs A B A  B A (A  B) A (A  B)  B F F T F T F T T F T T F F F T T T T T T Direct Proof using a truth table Prove that A  (A  B)   B is a contradiction. A  (A  B)   B = A  (A  B)   B = (A A)  (A  B)   B using distributive law. = F  (A  B)   B = (A  B)   B = A  (B   B) using associative law = A  F = F Contradiction. Hence B must be true. Proof by contradiction

  8. An Important Theorem Among the four statements p  q statement q  p converse p  q inverse q  p contra-positive 1. The statement and its contra-positive are equivalent. 2. The converse and inverse are equivalent. 3. No other pairs in the statements given above are equivalent. Proof:Make a truth table to see that 1 and 2 are tautologies.

  9. Prove that is not a rational number. (Example 1.7, page 13) Proof: Assume is a rational number. Let m and n be two integers, with no common factor, such that This shows that is even. Proof By Contradiction 1 1 Or Since is a multiple of 2, it is an even number. Therefore, n is even must be of the form 2k for some integer k.

  10. We have concluded that both and are even, therefore, n and m are also even They must have a common factor. This contradicts our assumption. Hence is not a rational number. Proof Continued …

  11. Proof of Proposition 1

More Related